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Pulling Out Rules

Module by: Interactive Mathematics Program

Intent

This activity gives students more opportunities to find and express rules for In-Out tables, both in words and symbolically, and to use an In-Out table as a problem-solving tool.

Mathematics

  • This activity raises the several mathematical issues.
  • Finding rules for In-Out tables.
  • Developing symbol sense. For example, “The Out is 2 times the In, then add 3,” Out = 2 In + 3, Out = 2 x + 3, and y = 2x +3 are equivalent and increasingly abstract ways to use symbols to summarize the rule for Question 1a.
  • Confronting the idea that the number of data points in a table affects the number of rules that will explain all the data.
  • Using a function as a mathematical model of a quantitative situation, and then using the model to solve a problem related to that situation.
  • Introducing the terms variable, algebraic expression, coefficient, and constant term.

Progression

Students are first asked to find rules for In-Out tables that each contain four pairs of numbers. Then they are asked to generate many possible rules that fit tables with only one or two pairs of numbers. Finally, they are presented with a problem for which an In-Out table is a helpful solution tool.

Approximate Time

25 minutes for activity

20 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Doing the Activity

Tell students that in this activity they will look for rules for more In-Out tables.

Discussing and Debriefing the Activity

Students should begin by asking questions of and comparing results with their group members. During this time, ask for some groups to prepare presentations for one of the tables.

During presentations, encourage students to talk about the strategies they used to find their rules. Ensure that rules are presented as complete sentences, and record these sentences on the board.

Now that students have worked with a variety of In-Out tables, it may be valuable to let them share techniques that they have found for finding rules. As they do, help them to clarify their own thinking and encourage them to challenge each other to explain their ideas clearly.

Make sure a variety of rules for each table are shared. In Question 2, with only one data point to explain, students can find an infinite number of possible rules. If an In-Out table contains two points, as in Question 3, there are still infinitely many possible rules, but there is now only one linear rule that will work.

Question 4a asks students to create an In-Out table to analyze a real-life situation.

Number of volunteers Number of bags of weeds pulled
1 2
2 5
3 8

Some students will answer Question 4b by extending the table using the pattern “Add 3 bags for each additional person” until they reach an output that is large enough. Some might find and employ a rule that fits the table Number of bags = 3 • (Number of volunteers) + 1.

Using either approach, students will discover that 30 does not appear as an Out if they stick to whole numbers for the In. So what should the supervisor do? To address this question, students must make sense of their work so far in light of the problem context. Their ideas could include these.

•The supervisor should “play it safe” by getting 11 volunteers, taking into account that the job might actually involve more than 30 bags of weeds and that some volunteers might work faster than others.

•The supervisor should get 10 volunteers and either make them work extra hard or join in the work as needed to finish the job.

There is no right answer; each of these suggestions (and other possible ideas) makes sense in this context.

Begin moving students from rules expressed in words to rules expressed symbolically. The discussion may be richer if this is a two-operation rule, such as “Get the Out by tripling the In and then adding 1.”

Have students find the Out that goes with each of several In values for their rules. For the rule just stated, the table might look something like this.

In Out
1 4
3 10
6 19
5 16

Help students record the rules as algebraic expressions. Students already know symbols for the numbers and operations involved, and most will have used symbols to replace unknown quantities. Remind them how much they know already about writing algebraic expressions.

With a few volunteered responses, students should arrive at an algebraic expression that all agree finishes the sentence. Record the expression in the In-Out table.

In Out
1 4
3 10
6 19
5 16
t 3∑t + 1

Below are a few additional ideas about algebraic notation and terminology that you can either elicit from students or simply present, using the context of the table just discussed.

  • It is conventional to abbreviate 3∑t + 1 as 3t + 1.
  • It is acceptable to write t ∑3, unusual to write t 3, and most common to write 3t. Emphasize that omitting the multiplication sign is simply a convention of notation—that is, an agreement among mathematicians to write things a certain way. There is nothing inherently wrong about using a multiplication sign between a number and a variable or about placing the variable in front of the coefficient—it’s just not usually done that way.
  • The letter t is a variable. Rather than formally defining the term, you might just say that a variable is a letter that is being used to represent a general case.
  • 3t + 1 is an algebraic expression. A number used to multiply a variable, such as 3 in the expression 3t + 1, is a coefficient.
  • A number by itself that is added or subtracted in such an expression, such as 1 in the expression 3t + 1, is a constant term.

After introducing this terminology and notation, you might ask students to express some of the other rules they found for the tables in Questions 2 and 3 in algebraic form. Using a variety of letters for the inputs emphasizes that the particular letter chosen has no significance.

Key Questions

What rules have we found for this table? Are these rules really different? That is, would they lead to different tables? Or are they different ways of stating the same rule?

What should the supervisor do?

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